422 research outputs found
On the distribution of conjugacy classes between the cosets of a finite group in a cyclic extension
Let G be a finite group and H a normal subgroup such that G/H is cyclic.
Given a conjugacy class g^G of G we define its centralizing subgroup to be
HC_G(g). Let K be such that H\le K\le G. We show that the G-conjugacy classes
contained in K whose centralizing subgroup is K, are equally distributed
between the cosets of H in K. The proof of this result is entirely elementary.
As an application we find expressions for the number of conjugacy classes of K
under its own action, in terms of quantities relating only to the action of G.Comment: 12 page
On exceptional groups of order p^5
A finite group G is exceptional if it has a quotient Q whose minimal faithful
permutation degree is greater than that of G. We say that Q is a distinguished
quotient.
The smallest examples of exceptional p-groups have order p^5. For an odd
prime p, we classify all pairs (G,Q) where G has order p^5 and Q is a
distinguished quotient. (The case p=2 has already been treated by Easdown and
Praeger.) We establish the striking asymptotic result that as p increases, the
proportion of groups of order p^5 with at least one exceptional quotient tends
to 1/2.Comment: 23 page
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